This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra $sl_n$. The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.
Keywords: category, functor, abelian categorification, braid group, Hecke algebra, Weyl algebra
2000 MSC: 18A22; 18A25; 16D90; 18E10; 18E15; 20C08
Theory and Applications of Categories,
Vol. 22, 2009,
No. 19, pp 479-508.